In consideration of Orbital Angular Momentum (OAM) potentialities of increasing transmission capacity and since radio frequency (RF) spectrum shortage problem is deeply felt in radio communications sector, recently a lot of experimental studies have been carried out on the use of OAM states, or modes, at RF (also known as radio vortices) in order to try to enhance RF spectrum reuse.
In this connection, reference may, for example, be made to:                Mohammadi S. M. et al., “Orbital Angular Momentum in Radio—A System Study”, IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, IEEE SERVICE CENTER, PISCATAWAY, N.J., US, vol. 58, no. 2, 1 Feb. 2010, pages 565-572, which shows that standard antennas arranged in circular arrays can be used to generate RF beams carrying OAM;        Tamburini F. et al., “Encoding many channels in the same frequency through radio Vorticity: first experimental test”, arXiv.org, 12 Jul. 2011, Ithaca, N.Y., USA, which experimentally shows that it is possible to propagate and use the properties of twisted non-monochromatic incoherent radio waves to simultaneously transmit several radio channels on one and the same frequency by encoding them in different (and, thence, orthogonal) OAM states (even without using polarization or dense coding techniques);        GB 2 410 130 A, which discloses a planar phased array antenna for transmitting and receiving OAM radio vortex modes, which antenna comprises a circular array of cavity backed axial mode spiral antenna elements whose phase is controlled such that the phase of each antenna element changes sequentially about the array; and        WO 2012/084039 A1, which discloses a transmit antenna arrangement comprising N antenna elements arranged along a circumference with an angular separation of α degrees between neighboring antenna elements, the antenna arrangement comprising an OAM encoder arranged to receive N input signals for transmission, indexed from M=−(N−1)/2 up to M=(N−1)/2 for odd N and from M=−(N−2)/2 up to N/2 for even N; the OAM encoder connecting each input signal to each antenna element and giving each input signal M at each antenna element a phase shift of M*α relative to the phase of the same input signal M at an adjacent antenna element; wherein two or more antenna elements are directional, have their directivity in the same direction, and have an antenna aperture higher than, or equal to, 5λ, where λ is the wavelength of the N input signals.        
From a mathematical perspective, the transmission of an OAM mode (or state) at a single RF (i.e., by using a pure tone) implies that the electrical field on the radiating aperture can be represented as:F(ρ,ϕ)=F(ρ)ejkϕ,where ρ and ϕ are the cylindrical coordinates on the radiating aperture, j is the imaginary unit, and k is a positive or negative integer.
The radiated field can be represented in the far zone as:
            E      ⁡              (                  ϑ          ,          φ                )              =                  1        R            ⁢                        ∫          ∫                S            ⁢              F        ⁡                  (                      ρ            ,            ϕ                    )                    ⁢              e                              -            j                    ⁢                                          ⁢          2          ⁢                                          ⁢          π          ⁢                      ρ            λ                    ⁢                      sin            ⁡                          (              ϑ              )                                ⁢                      cos            ⁡                          (                              φ                -                ϕ                            )                                          ⁢      ρ      ⁢                          ⁢      d      ⁢                          ⁢      ρ      ⁢                          ⁢      d      ⁢                          ⁢      ϕ        ,where ϑ and φ are the spherical coordinates in the far field, R denotes the radius of the sphere centered on the radiating aperture, S denotes the integration surface used at reception side, and λ denotes the wavelength used.
As is known, due to intrinsic characteristics of OAM, an OAM mode transmitted at a single RF (i.e., by using a pure tone) is affected by a phase singularity which creates a null at the bore-sight direction, thereby resulting thatE(0,0)=0.
In order for said phase singularity to be compensated, the integration surface S used at reception side should be sized so as to include the crown-shaped peak generated by the OAM mode.
In particular, the integration surface S used at reception side should be different for each OAM mode and, considering the sampling theorem applied to the radiating antenna, should have an area given by:
                    Δ        ⁢                                  ⁢        S            -              Δ        ⁢                                  ⁢        Ω        ⁢                                  ⁢                  R          2                      =          2      ⁢                        (                                    λ              D                        ⁢            R                    )                2              ,where D denotes the diameter of the radiating antenna.
Therefore, the price to be paid with pure OAM modes transmitted by using pure tones (i.e., single radiofrequencies) is that the dimensions of the equivalent receiving antenna depend on the distance R from, and on the diameter D of, the transmitting antenna.
This solution is impractical for many applications, especially for satellite communications where the aperture efficiency and the size of the antennas are very critical issues. For example, in geostationary-satellite-based communications in Ka band, for a ground antenna having a diameter D of about 9 m, the diameter of the receiving ring on board the geostationary satellite should be of the order of 50 km, thereby resulting impractical.
Thence, in view of the foregoing, the main criticality in using radio vorticity in practical systems is that the orthogonality between OAM modes depends on the size of antennas, on the distance between the transmitting and receiving antennas, and on the need for the receiving antenna to operate as an interferometer basis. These constraints result in OAM-based radio communication systems which are inefficient and unusable for very long distances, such as the ones involved in satellite communications.
Moreover, further criticalities in the use of radio vorticity for satellite communications are represented by the need of an extremely accurate mutual pointing of the transmitting and receiving antennas, and by the unfeasibility of the geometry for Earth-satellite configurations due to the criticality of the positioning of the receiving antennas (or of the receiving antenna elements).
A solution to the aforesaid technical problems is provided in Applicant's International application WO 2014/016655 A1 (whose content is herewith enclosed by reference), that concerns a multidimensional space modulation technique for transmitting and receiving radio vortices at frequencies ranging from a few kHz to hundreds of GHz. Specifically, the multidimensional space modulation technique according to WO 2014/016655 A1 allows to transmit and receive orthogonal RF OAM modes in one and the same direction (i.e., the bore-sight direction) and to overcome, at the same time, the aforesaid technical problems caused by OAM phase singularity at the bore-sight direction, thereby allowing the use of radio vortices also for long-distance radio communications, such as satellite communications.
In particular, the multidimensional space modulation according to WO 2014/016655 A1 is actually a phase modulation applied to signals to be transmitted at RF such that to result in orthogonal radio vortices along the bore-sight direction. Therefore, the modulation according to WO 2014/016655 A1 is conveniently called multidimensional space modulation because it allows orthogonal RF OAM modes to be transmitted and received in one and the same direction, namely the bore-sight direction, wherein each OAM mode represents a specific space channel along the bore-sight direction, which specific space channel is orthogonal to all the other space channels represented by the other OAM modes.
In order for the multidimensional space modulation according to WO 2014/016655 A1 to be better understood, attention is drawn, by way of example, to the fact that, as is known, a twisted RF signal having, or carrying, the OAM mode m=+1 (where m, as is known, is called topological charge) is characterized by only one clockwise rotation of 360° of the Poynting vector around the propagation axis per period T and, thence, it can be generated by transmitting, for example by means of four ring-arranged transmitting antenna elements, RF signals associated with phases of 0°, 90°, 180°, and 270° clockwise distributed among said four ring-arranged transmitting antenna elements. Instead, WO 2014/016655 A1 proves that it is possible and convenient, in order to transmit at RF the OAM mode m=+1 and, at the same time, to solve the problem caused by OAM phase singularity at the bore-sight direction, to exploit only one antenna transmitting the four different phases 0°, 90°, 180°, and 270° at different times (or at different frequencies) with a time step of T′=T/4. This possibility increases the efficiency of the transmitting and receiving configuration, which can work regardless of the elementary antenna element spacing in an antenna array.
From a conceptual perspective, according to WO 2014/016655 A1, in order to manage OAM rotation, namely in order to control the speed of rotation of an RF OAM mode about the bore-sight direction, a supplementary phase modulation is introduced, which leaves only a residue of the OAM twist and keeps the OAM signature in a limited bandwidth. This residual rotation achieved by means of the supplementary phase modulation allows a signal having a proper bandwidth to be orthogonal to another signal having a different rotation (multiple of the minimum one). Therefore, an RF twisted wave can be transmitted by means of a modulated waveform and can be received by an antenna operating in the complex conjugated mode. The received signal is equal to the transmitted one, except for standard attenuation and transmission and reception gains. The bandwidth increase does not prevent the transmission of plane waves (i.e., the OAM mode m=0), but limits the number of OAM modes at different frequencies in the available bandwidth. The multidimensional space modulation according to WO 2014/016655 A1 allows to use a standard antenna in place of a phased array antenna, since the used signals are natively orthogonal.
In detail, WO 2014/016655 A1 discloses a device for generating OAM modes for radio communications, which device is designed to receive one or more input digital signals, each of which:                has a respective sampling period, which is a respective multiple of a given sampling period; and        occupies a frequency bandwidth, which is a respective fraction of a given available frequency bandwidth.        
The device for generating OAM modes according to WO 2014/016655 A1 is:                operable to                    apply, to each input digital signal, a respective space modulation associated with a respective OAM mode having a respective topological charge to generate a corresponding modulated digital signal carrying said respective OAM mode, and            provide an output digital signal based on the modulated digital signal(s); and                        configured to apply, to each input digital signal, the respective space modulation by interpolating said input digital signal and phase-modulating the interpolated input digital signal so as to generate a corresponding phase-modulated digital signal carrying said respective OAM mode, having the given sampling period, and occupying the given available frequency bandwidth.        
In particular, the device for generating OAM modes according to WO 2014/016655 A1 is configured to apply, to each input digital signal, the respective space modulation by:                digitally interpolating said input digital signal thereby generating a corresponding digitally-interpolated signal having the given sampling period; and        phase-modulating the corresponding digitally-interpolated signal on the basis of digital phase shifts related to the respective OAM mode so as to generate the corresponding phase-modulated digital signal.        
For example, in order to generate OAM mode +1, WO 2014/016655 A1 teaches to:                digitally interpolate an input digital signal (having a sampling period equal to 4T0, where T0 denotes the given sampling period) by outputting, for each digital sample of said digital signal, four corresponding digital samples with time step (i.e., time distance) T0, thereby generating a corresponding digitally-interpolated signal having the given sampling period T0;        apply, to each set of four digital samples obtained by means of the digital interpolation, digital phase shifts related to the OAM mode +1 (namely, digital phase shifts related to phase values 0, π/2, π and 3π/2) so as to generate a corresponding set of four phase-shifted digital samples, which corresponding set of four phase-shifted digital samples carries the OAM mode +1; and        combine the sets of four phase-shifted digital samples into a single phase-modulated digital signal carrying the OAM mode +1, having the given sampling period T0, and occupying the given available frequency bandwidth.        
Accordingly, in order to generate OAM mode −1, WO 2014/016655 A1 teaches to:                digitally interpolate an input digital signal (having a sampling period equal to 4T0, where T0 denotes the given sampling period) by outputting, for each digital sample of said digital signal, four corresponding digital samples with time step (i.e., time distance) T0, thereby generating a corresponding digitally-interpolated signal having the given sampling period T0;        apply, to each set of four digital samples obtained by means of the digital interpolation, digital phase shifts related to the OAM mode −1 (namely, digital phase shifts related to phase values 0, −3π/2, −π and −π/2) so as to generate a corresponding set of four phase-shifted digital samples, which corresponding set of four phase-shifted digital samples carries the OAM mode −1; and        combine the sets of four phase-shifted digital samples into a single phase-modulated digital signal carrying the OAM mode −1, having the given sampling period T0, and occupying the given available frequency bandwidth.        
The generation of higher-order OAM modes (i.e., with |m|>1, where m denotes the topological charge of the OAM mode considered) according to WO 2014/016655 A1 is performed, mutatis mutandis, conceptually in the same way as the generation of OAM modes ±1 previously described.
Additionally, Applicant's International applications WO 2015/067987 A1 and WO 2015/068036 A1 (whose contents are herewith enclosed by reference) disclose, both, the feasibility of increasing transmission capacity at RF (including frequencies from a few kHz to hundreds of GHz) by exploiting a proper approximation in time domain of the Hilbert transform of digital analytical signals, wherein said approximation of the Hilbert transform is obtained by exploiting time twisted waves.
Instead, Applicant's International applications WO 2015/189653 A1 and WO 2015/189704 A2 (whose contents are herewith enclosed by reference) teach, by exploiting duality between time and frequency, to use also a twisted-wave-based approximation of the Hilbert transform in frequency domain in order to increase transmission capacity.
In particular, as for time twisted waves, WO 2015/067987 A1 discloses a radio communications system comprising a transmitter and a receiver, wherein the transmitter is configured to:                generate or receive digital symbols having a given symbol rate associated with a corresponding symbol period;        generate, every S digital symbols generated/received (S being an integer higher than three), a respective multi-mode digital signal, which has a predefined time length shorter than S times the symbol period, which is sampled with a predefined sampling rate higher than the symbol rate, and which carries said S digital symbols by means of a plurality of orthogonal OAM modes comprising                    a main mode carrying P of said S digital symbols (P being an integer higher than zero and lower than S), and            one or more secondary modes carrying the other S-P digital symbols, each secondary mode being time-shifted by half the symbol period with respect to the main mode; and                        transmit a radio frequency signal carrying a sequence of the generated multi-mode digital signals.        
Moreover, the receiver of the radio communications system according to WO 2015/067987 A1 is configured to:                receive the radio frequency signal transmitted by the transmitter;        process the received radio frequency signal so as to obtain a corresponding incoming digital signal; and        extract, from successive, non-overlapped portions of the incoming digital signal sampled with the predefined sampling rate, the S digital symbols respectively carried by each incoming digital signal portion by means of the orthogonal OAM modes; wherein each of the successive, non-overlapped portions of the incoming digital signal has the predefined time length.        
More in detail, the transmitter of the radio communications system according to WO 2015/067987 A1 is configured to generate a multi-mode digital signal carrying S digital symbols by:                allocating P of the S digital symbols to the main mode by providing, for each of said P digital symbols, a corresponding complex value which represents said digital symbol and is related to the main mode;        allocating each of the other S-P digital symbols to a corresponding secondary mode by providing, for each of said S-P digital symbols, a corresponding complex value which represents said digital symbol and is related to the secondary mode to which said digital symbol is allocated;        computing, by using a predefined transmission matrix, M multi-mode complex values related to M successive time instants (M being an integer equal to, or higher than, S) which, within the predefined time length, are separated by half the symbol period, wherein the predefined transmission matrix relates                    the S complex values representing the S digital symbols and related to the OAM modes            to the M successive time instants            through complex coefficients each of which is related to a respective OAM mode and to a respective time instant; and                        generating a multi-mode digital signal having the predefined time length and sampled with the predefined sampling rate on the basis of the M multi-mode complex values computed.        
Moreover, the receiver of the radio communications system according to WO 2015/067987 A1 is configured to extract the S digital symbols carried by an incoming digital signal portion having the predefined time length and sampled with the predefined sampling rate by:                extracting, from said incoming digital signal portion, M multi-mode complex values related to M successive time instants which are, within the predefined time length, separated by half the symbol period;        computing, by using a reception matrix derived from the predefined transmission matrix through a generalized inversion technique (such as a pseudo-inverse technique), S complex values representing the S digital symbol carried by said incoming digital signal portion by means of the orthogonal OAM modes, wherein said reception matrix relates                    the M extracted multi-mode complex values related to the M successive time instants            to the S complex values to be computed            through complex coefficients each of which is related to a respective OAM mode and to a respective time instant; and                        determining the S digital symbols represented by the S complex values computed.        
Additionally, WO 2015/068036 A1 relates to a radio communications system and method based on time twisted waves. In particular, the radio communications method according to WO 2015/068036 A1 comprises carrying out, by a transmitter, the following steps:
a) generating or receiving digital symbols to be transmitted, said digital symbols having a given symbol rate associated with a corresponding symbol period;
b) generating, every S digital symbols generated/received (S being an integer higher than three), a corresponding multi-mode digital signal, which                has a predefined time length shorter than S times the symbol period,        has a predefined bandwidth larger than the Nyquist bandwidth corresponding to the given symbol rate, and        carries said S digital symbols by means of OAM modes comprising                    a main mode, that is an OAM mode with topological charge equal to zero and that carries P of said S digital symbols (P being an integer higher than zero and lower than S), and            one or more twisted modes carrying the other S-P digital symbols, wherein each twisted mode is an OAM mode with a respective topological charge different than zero and is time-shifted with respect to the main mode;                        
c) generating a multi-frame digital signal comprising successive, non-overlapped time frames, each of which has the predefined time length and carries a respective multi-mode digital signal generated; and
d) transmitting a radio frequency signal carrying the multi-frame digital signal.
Moreover, the radio communications method according to WO 2015/068036 A1 further comprises carrying out, by a receiver, the following steps:
e) receiving the radio frequency signal transmitted by the transmitter;
f) processing the received radio frequency signal so as to obtain a corresponding incoming digital signal;
g) performing on the basis of the incoming digital signal                carrier synchronization thereby recovering frequency and/or phase carrier used by the transmitter to generate the multi-mode digital signals,        clock synchronization thereby recovering the symbol rate and sampling time instants of the multi-mode digital signals generated by the transmitter, and        frame synchronization thereby detecting successive, non-overlapped portions of the incoming digital signal corresponding to the successive, non-overlapped time frames of the multi-frame digital signal generated by the transmitter; and        
h) extracting, on the basis of the carrier, clock and frame synchronizations performed, the S digital symbols respectively carried by each detected incoming digital signal portion by means of the OAM modes.
Instead, WO 2015/189653 A1 relates to a radio communications system and method with increased transmission capacity based on frequency twisted waves. In particular, the radio communications method according to WO 2015/189653 A1 comprises:                carrying out, by a transmitter, the steps of                    a) providing a digital time signal carrying digital symbols to be transmitted, and            b) transmitting a radio frequency signal carrying said digital time signal; and                        carrying out, by a receiver, the step of                    c) receiving the radio frequency signal transmitted by the transmitter,            d) processing the received radio frequency signal so as to obtain a corresponding incoming digital signal, and            e) extracting, from the incoming digital signal, the digital symbols carried by said incoming digital signal.                        
The radio communications method according to WO 2015/189653 A1 is characterized in that said digital time signal carrying the digital symbols to be transmitted results from an approximation of the Hilbert transform in frequency domain, which approximation is based on a frequency main mode and one or more frequency twisted modes, wherein said frequency main and twisted modes carry, each, respective digital symbols to be transmitted.
In detail, according to WO 2015/189653 A1, the digital time signal is time-limited, carries a limited sequence of digital symbols to be transmitted, and results from:                main mode frequency samples carrying respective digital symbols of said limited sequence via a frequency main mode; and        twisted mode frequency samples carrying the other digital symbols of said limited sequence via one or more frequency twisted modes, wherein each frequency twisted mode is an OAM mode that is orthogonal to the frequency main mode and to any other frequency twisted mode used.        
More in detail, the main mode frequency samples are at main mode frequencies spaced apart by a predetermined frequency spacing, and the twisted mode frequency samples comprise, for a frequency twisted mode, respective twisted mode frequency samples at corresponding twisted mode frequencies that:                are related to said frequency twisted mode;        are spaced apart by said predetermined frequency spacing; and        are different from the main mode frequencies.        
Additionally, also WO 2015/189704 A2 relates to a radio communications system and method with increased transmission capacity based on frequency twisted waves. In particular, the radio communications method according to WO 2015/189704 A2 comprises:                carrying out, by a transmitter, the steps of                    a) generating a digital time signal, that is time-limited, carries a limited sequence of digital symbols to be transmitted and results from an approximation of the Hilbert transform in frequency domain, which approximation is based on                            a frequency main mode, that is associated with an OAM mode with topological charge equal to zero, and that includes main mode frequency samples carrying respective digital symbols of said limited sequence via said OAM mode with topological charge equal to zero, and                one or more frequency twisted modes carrying the other digital symbols of said limited sequence, wherein each frequency twisted mode is associated with a corresponding OAM mode with a respective topological charge different than zero, and includes respective twisted mode frequency samples carrying one or more respective digital symbols of said limited sequence via said corresponding OAM mode with said respective topological charge different than zero; and                                    b) transmitting a radio frequency signal carrying the digital time signal generated; and                        carrying out, by a receiver, the step of                    c) receiving the radio frequency signal transmitted by the transmitter,            d) processing the received radio frequency signal so as to obtain a corresponding incoming digital signal, and            e) extracting, from the incoming digital signal, the digital symbols carried by said incoming digital signal.                        
In particular, according to WO 2015/189704 A2, for each frequency twisted mode, the respective twisted mode frequency samples are mutually phase-shifted on the basis of phase shifts related to the corresponding OAM mode with the respective topological charge different than zero.
Additionally, Applicant's International application WO 2015/189703 A2 relates to the use of frequency twisted waves to increase transmission capacity of:                in general, wireless communication systems based on Orthogonal Frequency-Division Multiplexing (OFDM) and/or Orthogonal Frequency-Division Multiple Access (OFDMA) and/or Single-Carrier Frequency-Division Multiple Access (SC-FDMA) and/or combinations/variants/developments of these technologies; and,        in particular, 4G (4th Generation) cellular networks based on Long Term Evolution (LTE) and/or LTE Advanced standards, future 5G (5th Generation) cellular networks, and also wireless communication systems based on Worldwide Interoperability for Microwave Access (WiMAX) standard.        
In particular, WO 2015/189703 A2 discloses a method for radio communications in a wireless communication system including one or more base stations and one or more user terminals, said method comprising performing a radio communication between a base station and a user terminal of said wireless communication system, wherein performing a radio communication includes transmitting, in a given time slot, first digital symbols by using a frequency-division technique, whereby the first digital symbols are carried by first frequency samples at respective sub-carriers, that are distributed over a predefined frequency band and belong to a given sub-carrier block.
The method according to WO 2015/189703 A2 is characterized in that performing a radio communication further includes transmitting, in said given time slot, also second digital symbols by means of one or more frequency twisted modes, wherein each frequency twisted mode carries a respective OAM mode with a respective topological charge different than zero by means of respective second frequency samples, that:                are phase-shifted with respect to each other on the basis of phase shifts related to said respective OAM mode;        carry one or more respective symbols of said second digital symbols via said respective OAM mode; and        are at respective frequencies, that are distributed over said predefined frequency band and are different than the sub-carriers belonging to said given sub-carrier block.        
In detail, according to WO 2015/189703 A2, the sub-carriers belonging to the given sub-carrier block are spaced apart by a predetermined frequency spacing, and each frequency twisted mode includes respective second frequency samples at respective frequencies that are spaced apart by said predefined frequency spacing and are distributed over said predefined frequency band between pairs of consecutive sub-carriers belonging to the given sub-carrier block.
More in detail, according to WO 2015/189703 A2, for each second digital symbol to be transmitted in the given time slot, a respective frequency twisted mode carries, by means of respective second frequency samples, said second digital symbol via a phase-modulation related to the OAM mode carried by said respective frequency twisted mode.